tl;dr: Subpar shooter with great story and visuals, but horrendous story-telling and grave collision issues.
If you remember BioShock 1, you remember the utter garbage the devs called "shooting mechanics". And if you're like me, the repetitive boring hacking minigame is what keeps you from giving it a shot again.
In this respect, BioShock Infinite improved a lot. The shooting mechanics actually work and are somewhat fun. Your limited to carrying two weapons (yay consoles!), but the feedback is better and you're not required to swap weapons to use vigors. However, there are balancing issues. For some reason you can look the sights of your weapon, but you'll suddenly suffer from heavy recoil, whereas recoil is almost non-existent when firing from the hip. New weapons you find later in the game are useless because your upgraded weapons are stronger – the game forces you to spend money on weapon upgrades, and that wouldn't be an issue for me if at least the standard RPG wasn't weaker than my pistol.
It's similar for the vigors. The second one you find is an overpowered fireball that does AoE damage, sets enemies on fire (damage over time) and even stuns them for that duration. It's the only one you'll effectively use, except for maybe the shock vigor (for stunning the two enemies immune to the fireball) and the fus ro dah in the final chapter.
Essentially, there are three types of enemies. The "headshot = instant kill" type, the bulletsponge type that survives 50 minigun shots in his face and the big ones that actually require a special tactic.
The bulletsponges aren't more difficult to beat than the first type, they just make the game more tedious. The only challenging enemy is the Handyman, which is basically a Big Daddy who keeps jumping into your face, and I've never found an effective tactic to stun or outmaneuver them. I played on hard difficulty, and the entire game was a joke except when meeting those.
Now the worst problem are the collisions. You get stuck everywhere. At doorframes, at corners of furniture, at invisible walls, even at curbsides and at hay lying on the ground. I noticed this the first time just in the beginning of the game when I arrived in Columbia and literally got stuck in a paper scroll lying on the ground. What in the world where the devs thinking when implemented those collision meshes? It destroys the flow of the gameplay and three times I had to restart at a checkpoint because I was stuck and couldn't get out.
Sometimes you want to hang on a hook and jump off the escape an enemy. Sometimes it simply doesn't work and the camera keeps resetting when you want to aim for a landing spot. Only near to the end of the game I finally realised that you have to stay still for a few seconds until it works, because there's an animation that simply resets when you move the mouse. Great gameplay flow!
BioShock Infinite builds a great atmosphere. And manages to destroy the immersion with the weirdest things. For instance, you can steal everything in the front of people's eyes – rob a store in front of its owner, empty a bar in front of the barkeeper and the patrons, empty the luggage of refugees fleeing the town. For some reason, there are two or three locations spread throughout the game where you suddenly get punished for stealing.
Then there are weird script events. Crowds spawned right in front of my eyes to block the way back. People just despawn from the street when the airship takes off. Streets are suddenly barricaded and filled with enemies 15 seconds after you left them.
When you arrive at Columbia, you receive a telegram saying "Don't pick #77". 15 minutes later you're at a raffle and pick a ball with the number 77, but Hooker just doesn't get the hint and that's where the trouble starts, lol wtf.
Now, as for the story: Throughout the game, there isn't any. You think you'll learn something new soon, but it doesn't happen. 90% of the story and all the explanations happen in the last hour of the game, during which it simply turns in a boring walking simulator. The devs completely failed in implenting the story of their game in the actual gameplay. You could literally watch the walking simulator ending on YouTube, skipping all the sub-par shooter gameplay before it and you'll know the whole story. Except for the audio tapes with background information, of course.
You probably know the audio tapes from BioShock 1. There is just a huge difference: In BioShock Infinite you have a companion and more enemies. Most times when I wanted to listen to the tapes, Elizabeth suddenly started a dialogue or I got into a fight. After a while I stopped listening to the tapes at all, because when the game actually wants you to stop playing the game (i.e. staying still and doing nothing) so you can pay attention to the story, it's simply horrendous story-telling. At this point I was already so annoyed by the repetitive gameplay and the horrible collision system that I stopped caring about everything anyway and only wanted to finally see the ending. Wasn't really worth it, considering that everything I got was a walking simulator.
Dear story writers, please start writing books and stop wasting your potential on sub-par videogames.